2012, 11(1): 83-96. doi: 10.3934/cpaa.2012.11.83

Existence of nontrivial steady states for populations structured with respect to space and a continuous trait

1. 

Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, 1040 Wien

2. 

ENS Cachan, CMLA, IUF & CNRS, PRES UniverSud, 61, Av. du Pdt Wilson, 94235 Cachan Cedex

3. 

MAPMO, Université d'Orléans, F-45067 Orléans Cedex, France

Received  January 2010 Revised  July 2010 Published  September 2011

We prove the existence of nontrivial steady states to reaction-diffusion equations with a continuous parameter appearing in selection/mutation/competition/migration models for populations, which are structured both with respect to space and a continuous trait.
Citation: Anton Arnold, Laurent Desvillettes, Céline Prévost. Existence of nontrivial steady states for populations structured with respect to space and a continuous trait. Communications on Pure & Applied Analysis, 2012, 11 (1) : 83-96. doi: 10.3934/cpaa.2012.11.83
References:
[1]

H. Brezis, "Analyse Fonctionnelle,", Masson, (1987).

[2]

F. Brezzi and G. Gilardi, "Fundamentals of P.D.E. for Numerical Analysis,", preprint n. 446 of Istituto di Analisi Numerica, (1984).

[3]

A. Calsina and S. Cuadrado, Asymptotic stability of equilibria of selection-mutation equations,, J. Math. Biol., 54 (2007), 489. doi: doi:10.1007/s00285-006-0056-4.

[4]

J. Carrillo, L. Desvillettes and K. Fellner, Exponential decay towards equilibrium for the inhomogeneous Aizenman-Bak model,, Commun. Math. Phys., 278 (2008), 433. doi: doi:10.1007/s00220-007-0404-2.

[5]

J. Carrillo, L. Desvillettes and K. Fellner, Fast-reaction limit for the inhomogeneous Aizenman-Bak model,, Kinetic and Related Models, 1 (2008), 127.

[6]

R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", vol 1. Physical Origins and Classical Methods, (1990).

[7]

L. Desvillettes, R. Ferrières and C. Prévost, "Infinite Dimensional Reaction-Diffusion for Population Dynamics,", preprint n. 2003-04 du CMLA, (): 2003.

[8]

L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On mutation-selection dynamics,, Commun. Math. Sc., 6 (2008), 729.

[9]

O. Diekmann, P.-E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach,, Theor. Popul. Biol., 67 (2005), 257. doi: doi:10.1016/j.tpb.2004.12.003.

[10]

P. Laurençot and S. Mischler, Global existence for the discrete diffusive coagulation-fragmentation equations in L1,, Rev. Mat. Iberoamericana, 18 (2002), 731.

[11]

P. Laurençot and S. Mischler, The continuous coagulation-fragmentation equations with diffusion,, Arch. Rational Mech. Anal., 162 (2002), 45.

[12]

G. Raoul, Local stability of evolutionary attractors for continuous structured populations,, to appear in Monatshefte f\, (2011).

[13]

F. Rothe, "Global Solutions of Reaction-Diffusion Systems,", Lecture Notes in Mathematics, (1072).

[14]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Second edition, (1994).

show all references

References:
[1]

H. Brezis, "Analyse Fonctionnelle,", Masson, (1987).

[2]

F. Brezzi and G. Gilardi, "Fundamentals of P.D.E. for Numerical Analysis,", preprint n. 446 of Istituto di Analisi Numerica, (1984).

[3]

A. Calsina and S. Cuadrado, Asymptotic stability of equilibria of selection-mutation equations,, J. Math. Biol., 54 (2007), 489. doi: doi:10.1007/s00285-006-0056-4.

[4]

J. Carrillo, L. Desvillettes and K. Fellner, Exponential decay towards equilibrium for the inhomogeneous Aizenman-Bak model,, Commun. Math. Phys., 278 (2008), 433. doi: doi:10.1007/s00220-007-0404-2.

[5]

J. Carrillo, L. Desvillettes and K. Fellner, Fast-reaction limit for the inhomogeneous Aizenman-Bak model,, Kinetic and Related Models, 1 (2008), 127.

[6]

R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", vol 1. Physical Origins and Classical Methods, (1990).

[7]

L. Desvillettes, R. Ferrières and C. Prévost, "Infinite Dimensional Reaction-Diffusion for Population Dynamics,", preprint n. 2003-04 du CMLA, (): 2003.

[8]

L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On mutation-selection dynamics,, Commun. Math. Sc., 6 (2008), 729.

[9]

O. Diekmann, P.-E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach,, Theor. Popul. Biol., 67 (2005), 257. doi: doi:10.1016/j.tpb.2004.12.003.

[10]

P. Laurençot and S. Mischler, Global existence for the discrete diffusive coagulation-fragmentation equations in L1,, Rev. Mat. Iberoamericana, 18 (2002), 731.

[11]

P. Laurençot and S. Mischler, The continuous coagulation-fragmentation equations with diffusion,, Arch. Rational Mech. Anal., 162 (2002), 45.

[12]

G. Raoul, Local stability of evolutionary attractors for continuous structured populations,, to appear in Monatshefte f\, (2011).

[13]

F. Rothe, "Global Solutions of Reaction-Diffusion Systems,", Lecture Notes in Mathematics, (1072).

[14]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Second edition, (1994).

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